Puzzle for July 28, 2020 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add F to both sides of eq.6: C + F + F = A – F + F which becomes C + 2×F = A In eq.3, replace A with C + 2×F: E – F = C + 2×F – C which becomes E – F = 2×F Add F to each side of the above equation: E – F + F = 2×F + F which makes E = 3×F
Hint #2
In eq.2, replace E with 3×F: D = 3×F + F which makes D = 4×F
Hint #3
In eq.5, substitute 4×F for D, and 3×F for E: A + C + F = 4×F + 3×F which becomes A + C + F = 7×F Subtract both C and F from each side of the equation above: A + C + F – C – F = 7×F – C – F which becomes eq.5a) A = 6×F – C
Hint #4
Substitute 6×F – C for A (from eq.5a) in eq.6: C + F = 6×F – C – F which becomes C + F = 5×F – C In the above equation, add C to both sides, and subtract F from both sides: C + F + C – F = 5×F – C + C – F which makes 2×C = 4×F Divide each side by 2: 2×C ÷ 2 = 4×F ÷ 2 which makes C = 2×F
Hint #5
Substitute 2×F for C in eq.5a: A = 6×F – 2×F which makes A = 4×F
Hint #6
Substitute 4×F for A in eq.4: B + F = 4×F – B In the equation above, subtract F from each side, and add B to each side: B + F – F + B = 4×F – B – F + B which makes 2×B = 3×F Divide both sides by 2: 2×B ÷ 2 = 3×F ÷ 2 which makes B = 1½×F
Solution
Substitute 4×F for A and D, 1½×F for B, 2×F for C, and 3×F for E in eq.1: 4×F + 1½×F + 2×F + 4×F + 3×F + F = 31 which simplifies to 15½×F = 31 Divide both sides of the above equation by 15½: 15½×F ÷ 15½ = 31 ÷ 15½ which means F = 2 making A = D = 4×F = 4 × 2 = 8 B = 1½×F = 1½ × 2 = 3 C = 2×F = 2 × 2 = 4 E = 3×F = 3 × 2 = 6 and ABCDEF = 834862